Before protons, neutrons and electrons were discovered, scientists could not use atomic number to sort the elements. They tried to classify them by arranging them in order of atomic weights instead. The early tables were incomplete, and if the strict order of atomic weights was followed, some elements ended up in the wrong group because their properties did not match the others in that column.
Mendeleev overcame some of these problems. He left gaps in his table for elements he thought had not yet been discovered, and in a few places he changed the order based on atomic weights so that an element lined up with others that were similar. He also predicted the properties of the missing elements.
Later, new elements were discovered and filled the gaps, and their properties matched his predictions. The predictions turned out to be correct, which supported his idea. Testing a prediction like this is how a new scientific idea is supported or refuted. Knowledge of isotopes later explained why ordering by atomic weight was not always correct. Isotopes are atoms of the same element with different masses, so the average atomic weight depends on the mix of isotopes, whereas the proton number does not.